Abstract

A determination of the aerosol particle size distribution function by using the particle spectrum extinction equation is an ill-posed integral equation of the first kind. To overcome this, we must incorporate regularization techniques. Most of the literature focuses on the Phillips–Twomey regularization or its variations. However, there are drawbacks for some applications in which the real aerosol distributions have large oscillations in a Junge-type distribution. The reason for this is that the scale matrix based on the norm of the second differences in the Phillips–Twomey regularization is too ill- conditioned to filter the large perturbations induced by the small algebraic spectrum of the kernel matrix and the additive noise. Therefore we reexamine the aerosol particle size distribution function retrieval problem and solve it in W 1,2 space. This setting is based on Sobolev's embedding theorem in which the approximate solution best simulates the true particle size distribution functions. For choosing the regularization parameters, we also develop an a posteriori parameter choice method, which is based on the discrepancy principle. Our numerical results are based on the remote sensing data measured by the CE318 sunphotometer in Jia Xiang County, Shan Dong Province, China, and are performed to show the feasibility of the proposed algorithms.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  26. J. Liu and C. H. Chen, "A study on remote-sensing inversion of atmospheric aerosol particle size distributions of Lanzhou City in winter," Plateau Meteorol. 23, 103-109 (2004), in Chinese.
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    [CrossRef] [PubMed]
  28. Y. S. Cheng, "Condensation detection and diffusion size separation techniques," in Aerosol Measurement: Principles, Techniques and Applications, K.Willeke and P.A.Baron, eds. (Van Nostrand Reinhold, 1993), pp. 427-451.
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    [CrossRef] [PubMed]
  30. E. O. Knutson and K. T. Whitby, "Aerosol classification by electrical mobility: apparatus, thoery and applications," J. Aerosol Sci. 6, 453-460 (1975).
    [CrossRef]
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  32. F. Q. Yan, H. L. Hu, and J. Zhou, "Measurements of number density distribution and imaginary part of refractive index of aerosol particles," Acta Opt. Sin. 23, 854-859 (2003).
  33. D. K. Killinger and N. Menyuk, "Laser remote sensing of the atmosphere," Science 235, 37-45 (1987).
    [CrossRef] [PubMed]
  34. G. E. Shaw, "Sun photometry," Bull. Am. Meteorol. Soc. 64, 4-10 (1983).
    [CrossRef]
  35. W. C. Hinds, Aerosol Technology. Properties Behavior and Measurement of Airborne Particles (Wiley, 1982).
  36. K. Willeke and P. A. Baron eds., Aerosol Measurement: Principles, Techniques and Applications (Van Nostrand Reinhold, 1993).
  37. P. H. McMurry, "A review of atmospheric aerosol measurements," Atmos. Environ. 34, 1959-1999 (2000).
    [CrossRef]
  38. A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31 (Suppl. 1), 767-768 (2000).
    [CrossRef]
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  41. T. Y. Xiao, Sh. G. Yu, and Y. F. Wang, Numerical Methods for the Solution of Inverse Problems (Science Press, 2003).
  42. Y. F. Wang and T. Y. Xiao, "Fast realization algorithms for determining regularization parameters in linear inverse problems," Inverse Probl. 17, 281-291 (2001).
    [CrossRef]

2004

P. Demokritou, S. J. Lee, S. T. Ferguson, and P. Koutrakis, "A compact multistage (cascade) impactor for the characterization of atmospheric aerosols," J. Aerosol Sci. 35, 281-299 (2004).
[CrossRef]

J. Liu and C. H. Chen, "A study on remote-sensing inversion of atmospheric aerosol particle size distributions of Lanzhou City in winter," Plateau Meteorol. 23, 103-109 (2004), in Chinese.

2003

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

F. Q. Yan, H. L. Hu, and J. Zhou, "Measurements of number density distribution and imaginary part of refractive index of aerosol particles," Acta Opt. Sin. 23, 854-859 (2003).

2002

J. T. Mao, C. C. Li, J. H. Zhang, X. Y. Liu, and K. H. L. Alexis, "The comparison of remote sensing aerosol optical depth from MODIS data and ground Sun-photometer observations," J. Appl. Meteorol. Sci. 13, 127-135 (2002), in Chinese.

2001

Y. F. Wang and T. Y. Xiao, "Fast realization algorithms for determining regularization parameters in linear inverse problems," Inverse Probl. 17, 281-291 (2001).
[CrossRef]

C. Bockmann, "Hybrid regularization method for the ill-posed inversion of multiwave-length lidar data in the retrieval of aerosol size distributions," Appl. Opt. 40, 1329-1342 (2001).
[CrossRef]

2000

P. H. McMurry, "A review of atmospheric aerosol measurements," Atmos. Environ. 34, 1959-1999 (2000).
[CrossRef]

A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31 (Suppl. 1), 767-768 (2000).
[CrossRef]

1999

1996

1994

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

J. Heintzenberg, "Properties of lognormal particle size distributions," Aerosol Sci. Technol. 21, 46-48 (1994).
[CrossRef]

G. Ramachandran, D. Leith, and L. Todd, "Extraction of spatial aerosol distribution from multispectral light extinction measurements with computed tomography," J. Opt. Soc. Am. A 11, 144-154 (1994).
[CrossRef]

M. Weindisch and W. von Hoyningen-Huen, "Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements," Atmos. Environ. 28, 785-792 (1994).
[CrossRef]

1992

G. Ramachandran and D. Leith, "Extraction of aerosol-size distribution from multispectral light extinction data," Aerosol Sci. Technol. 17, 303-325 (1992).
[CrossRef]

1987

D. K. Killinger and N. Menyuk, "Laser remote sensing of the atmosphere," Science 235, 37-45 (1987).
[CrossRef] [PubMed]

1983

G. E. Shaw, "Sun photometry," Bull. Am. Meteorol. Soc. 64, 4-10 (1983).
[CrossRef]

1980

B. L. Preidecker, "Bacterial mutagenicity of particulates from Houston air," Environ. Mutagen. 2, 75-83 (1980).
[CrossRef] [PubMed]

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic Lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

1979

1978

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

1975

S. Twomey, "Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions," J. Comput. Phys. 18, 188-200 (1975).
[CrossRef]

D. Sinclair and G. S. Hoopes, "A novel form of diffusion battery," Am. Ind. Hyg. Assoc. J. 36, 39-42 (1975).
[CrossRef] [PubMed]

E. O. Knutson and K. T. Whitby, "Aerosol classification by electrical mobility: apparatus, thoery and applications," J. Aerosol Sci. 6, 453-460 (1975).
[CrossRef]

1974

C. N. Davies, "Size distribution of atmospheric aerosol," J. Aerosol Sci. 5, 293-300 (1974).
[CrossRef]

1971

1969

1963

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 1097-101 (1963).
[CrossRef]

1962

D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach. 9, 84-97 (1962).
[CrossRef]

1961

Ackerman, S. A.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

Alexis, K. H. L.

J. T. Mao, C. C. Li, J. H. Zhang, X. Y. Liu, and K. H. L. Alexis, "The comparison of remote sensing aerosol optical depth from MODIS data and ground Sun-photometer observations," J. Appl. Meteorol. Sci. 13, 127-135 (2002), in Chinese.

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of III-Posed Problems (Wiley, 1977).

Baron, P. A.

K. Willeke and P. A. Baron eds., Aerosol Measurement: Principles, Techniques and Applications (Van Nostrand Reinhold, 1993).

Baum, B. A.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

Bockmann, C.

Bohren, G. F.

G. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Byrne, D. M.

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic Lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

Chen, C. H.

J. Liu and C. H. Chen, "A study on remote-sensing inversion of atmospheric aerosol particle size distributions of Lanzhou City in winter," Plateau Meteorol. 23, 103-109 (2004), in Chinese.

Cheng, Y. S.

Y. S. Cheng, "Condensation detection and diffusion size separation techniques," in Aerosol Measurement: Principles, Techniques and Applications, K.Willeke and P.A.Baron, eds. (Van Nostrand Reinhold, 1993), pp. 427-451.

Cox, K.

T. Nguyen and K. Cox, "A method for the determination of aerosol particle distributions from light extinction data," in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, 1989), pp. 330-330.

Curcio, J. A.

Davies, C. N.

C. N. Davies, "Size distribution of atmospheric aerosol," J. Aerosol Sci. 5, 293-300 (1974).
[CrossRef]

Demokritou, P.

P. Demokritou, S. J. Lee, S. T. Ferguson, and P. Koutrakis, "A compact multistage (cascade) impactor for the characterization of atmospheric aerosols," J. Aerosol Sci. 35, 281-299 (2004).
[CrossRef]

Ferguson, S. T.

P. Demokritou, S. J. Lee, S. T. Ferguson, and P. Koutrakis, "A compact multistage (cascade) impactor for the characterization of atmospheric aerosols," J. Aerosol Sci. 35, 281-299 (2004).
[CrossRef]

Frey, R. A.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

Gebhart, J.

J. Gebhart, "Optical direct-reading techniques: light intensity systems," in Aerosol Measurement: Principles, Techniques and Applications, K.Willeke and P.A.Baron, eds. (Van Nostrand Reinhold, 1993), pp. 313-344.

Grassl, H.

Heintzenberg, J.

J. Heintzenberg, "Properties of lognormal particle size distributions," Aerosol Sci. Technol. 21, 46-48 (1994).
[CrossRef]

Herman, B. M.

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic Lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

Hinds, W. C.

W. C. Hinds, Aerosol Technology. Properties Behavior and Measurement of Airborne Particles (Wiley, 1982).

Hoopes, G. S.

D. Sinclair and G. S. Hoopes, "A novel form of diffusion battery," Am. Ind. Hyg. Assoc. J. 36, 39-42 (1975).
[CrossRef] [PubMed]

Hu, H. L.

F. Q. Yan, H. L. Hu, and J. Zhou, "Measurements of number density distribution and imaginary part of refractive index of aerosol particles," Acta Opt. Sin. 23, 854-859 (2003).

Hu, T.

Huffman, D. R.

G. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Kaipio, J. P.

A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31 (Suppl. 1), 767-768 (2000).
[CrossRef]

Killinger, D. K.

D. K. Killinger and N. Menyuk, "Laser remote sensing of the atmosphere," Science 235, 37-45 (1987).
[CrossRef] [PubMed]

King, D. M.

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic Lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

King, M. D.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

Knutson, E. O.

E. O. Knutson and K. T. Whitby, "Aerosol classification by electrical mobility: apparatus, thoery and applications," J. Aerosol Sci. 6, 453-460 (1975).
[CrossRef]

Koutrakis, P.

P. Demokritou, S. J. Lee, S. T. Ferguson, and P. Koutrakis, "A compact multistage (cascade) impactor for the characterization of atmospheric aerosols," J. Aerosol Sci. 35, 281-299 (2004).
[CrossRef]

Lee, S. J.

P. Demokritou, S. J. Lee, S. T. Ferguson, and P. Koutrakis, "A compact multistage (cascade) impactor for the characterization of atmospheric aerosols," J. Aerosol Sci. 35, 281-299 (2004).
[CrossRef]

Leith, D.

G. Ramachandran, D. Leith, and L. Todd, "Extraction of spatial aerosol distribution from multispectral light extinction measurements with computed tomography," J. Opt. Soc. Am. A 11, 144-154 (1994).
[CrossRef]

G. Ramachandran and D. Leith, "Extraction of aerosol-size distribution from multispectral light extinction data," Aerosol Sci. Technol. 17, 303-325 (1992).
[CrossRef]

Li, C. C.

J. T. Mao, C. C. Li, J. H. Zhang, X. Y. Liu, and K. H. L. Alexis, "The comparison of remote sensing aerosol optical depth from MODIS data and ground Sun-photometer observations," J. Appl. Meteorol. Sci. 13, 127-135 (2002), in Chinese.

Liu, J.

J. Liu and C. H. Chen, "A study on remote-sensing inversion of atmospheric aerosol particle size distributions of Lanzhou City in winter," Plateau Meteorol. 23, 103-109 (2004), in Chinese.

Liu, X. Y.

J. T. Mao, C. C. Li, J. H. Zhang, X. Y. Liu, and K. H. L. Alexis, "The comparison of remote sensing aerosol optical depth from MODIS data and ground Sun-photometer observations," J. Appl. Meteorol. Sci. 13, 127-135 (2002), in Chinese.

Lu, Y.

Lumme, K.

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

Mao, J. T.

J. T. Mao, C. C. Li, J. H. Zhang, X. Y. Liu, and K. H. L. Alexis, "The comparison of remote sensing aerosol optical depth from MODIS data and ground Sun-photometer observations," J. Appl. Meteorol. Sci. 13, 127-135 (2002), in Chinese.

McCartney, G. J.

G. J. McCartney, Optics of the Atmosphere (Wiley, 1976).

McMurry, P. H.

P. H. McMurry, "A review of atmospheric aerosol measurements," Atmos. Environ. 34, 1959-1999 (2000).
[CrossRef]

Menyuk, N.

D. K. Killinger and N. Menyuk, "Laser remote sensing of the atmosphere," Science 235, 37-45 (1987).
[CrossRef] [PubMed]

Menzel, W. P.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

Nguyen, T.

T. Nguyen and K. Cox, "A method for the determination of aerosol particle distributions from light extinction data," in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, 1989), pp. 330-330.

Phillips, D. L.

D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind," J. Assoc. Comput. Mach. 9, 84-97 (1962).
[CrossRef]

Platnick, S.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

Preidecker, B. L.

B. L. Preidecker, "Bacterial mutagenicity of particulates from Houston air," Environ. Mutagen. 2, 75-83 (1980).
[CrossRef] [PubMed]

Rahola, J.

K. Lumme and J. Rahola, "Light scattering by porous dust particles in the discrete-dipole approximation," Astrophys. J. 425, 653-667 (1994).
[CrossRef]

Ramachandran, G.

G. Ramachandran, D. Leith, and L. Todd, "Extraction of spatial aerosol distribution from multispectral light extinction measurements with computed tomography," J. Opt. Soc. Am. A 11, 144-154 (1994).
[CrossRef]

G. Ramachandran and D. Leith, "Extraction of aerosol-size distribution from multispectral light extinction data," Aerosol Sci. Technol. 17, 303-325 (1992).
[CrossRef]

Reagan, A. J.

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic Lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

Reagan, J. A.

M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, "Aerosol size distributions obtained by inversion of spectral optical depth measurements," J. Atmos. Sci. 35, 2153-2167 (1978).
[CrossRef]

Riédi, J. C.

S. Platnick, M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riédi, and R. A. Frey, "The MODIS cloud products: algorithms and examples from Terra," IEEE Trans. Geosci. Remote Sens. 41, 459-473 (2003).
[CrossRef]

Shaw, G. E.

Shifrin, K. S.

Sinclair, D.

D. Sinclair and G. S. Hoopes, "A novel form of diffusion battery," Am. Ind. Hyg. Assoc. J. 36, 39-42 (1975).
[CrossRef] [PubMed]

Spinhirne, J. D.

A. J. Reagan, D. M. Byrne, D. M. King, J. D. Spinhirne, and B. M. Herman, "Determination of complex refractive index and size distribution of atmospheric particles from bistatic-monostatic Lidar and solar radiometer measurements," J. Geophys. Res. 85, 1591-1599 (1980).
[CrossRef]

Tanaka, M.

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of III-Posed Problems (Wiley, 1977).

Todd, L.

Twomey, S.

S. Twomey, "Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions," J. Comput. Phys. 18, 188-200 (1975).
[CrossRef]

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 1097-101 (1963).
[CrossRef]

S. Twomey, Atmospheric Aerosols (Elsevier, 1977).

von Hoyningen-Huen, W.

M. Weindisch and W. von Hoyningen-Huen, "Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements," Atmos. Environ. 28, 785-792 (1994).
[CrossRef]

Voutilainenand, A.

A. Voutilainenand and J. P. Kaipio, "Statistical inversion of aerosol size distribution data," J. Aerosol Sci. 31 (Suppl. 1), 767-768 (2000).
[CrossRef]

Wang, S.

Wang, Y. F.

Y. F. Wang and T. Y. Xiao, "Fast realization algorithms for determining regularization parameters in linear inverse problems," Inverse Probl. 17, 281-291 (2001).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(Color online) Sunphotometer CE318.

Fig. 2
Fig. 2

Air-mass variation at local time from 2 May to 11 May 2005.

Fig. 3
Fig. 3

Variation of AOT on 2 May 2005.

Fig. 4
Fig. 4

Variation of AOT on 6 May 2005.

Fig. 5
Fig. 5

Variation of AOT on 8 May 2005.

Fig. 6
Fig. 6

Variation of AOT on 9 May 2005.

Fig. 7
Fig. 7

Particle size distribution on 2 May 2005.

Fig. 8
Fig. 8

Particle size distribution on 6 May 2005.

Fig. 9
Fig. 9

Particle size distribution on 8 May 2005.

Fig. 10
Fig. 10

Particle size distribution on 9 May 2005.

Tables (4)

Tables Icon

Table 1 Characteristics of CE318

Tables Icon

Table 2 Meteorological Data in the Jia Xiang Measurement Area a

Tables Icon

Table 3 Descriptions of the Abbreviations

Tables Icon

Table 4 Optimum Values α * of the Regularization Parameter for Every Daily Observation

Equations (174)

Equations on this page are rendered with MathJax. Learn more.

n ( r )
τ aero = β λ α
τ aero
τ aero
τ aero ( λ ) = 0 0 π r 2 Q ext ( r , λ , η ) n ( r , z ) d z d r ,
n ( r , z )
Q ext ( r , λ , η )
τ aero ( λ ) = 0 π r 2 Q ext ( r , λ , η ) n ( r ) d r ,
n ( r )
n ( r )
τ aero
n ( x )
{ o i } ( i = 1 , , m )
o i = a b K i ( x ) n ( x ) d x + ϵ i ( x = ln r ) ,
K i ( x )
n
( n j - n j - 1 ) 2 ]
n T H n
n
n T H n
ϵ i 2
n p ( i ) ( x ) = [ 1 + [ r p ( i - 1 ) - 1 ] K i ( x ) ] n p ( i - 1 ) ( x ) ,
r p ( i - 1 ) = o i K i ( x ) n p ( i - 1 ) ( x ) d x = K i ( x ) n ( x ) d x K i ( x ) n p ( i - 1 ) ( x ) d x .
( 300   nm < d 50 < 2500   nm
d 50
W 1 , 2
W 1 , 2
40   m
1016   hpa
68%
3 .3   m / s
3.1 m / s
33   cm
1020   nm
10   nm
870   nm
x a x b k ( x , y ) n ( y ) d y = o ( x ) ,
[ x a , x b ]
o ( x )
k ( x , y )
o i
n ( x )
x a x b k ( x , y ) n ( y ) d y + e ( x ) = o ( x ) + e ( x ) = d ( x ) ,
n ( x )
K : F O ,
K n = def 0 k ( r , λ , η ) n ( r ) d r = τ aero ,
k ( r , λ , η ) = π r 2 Q ex ( r , λ , η )
τ aero
K n + e = τ aero + e = d .
K n + e = d ,
K
l 2
min J lse [ n ] = 1 2 d K n l 2 2 .
J lse
K T K n K T d = 0.
K
K
cond ( K T K ) cond ( K ) ,
K n + e = d , e Δ
e
Q ( n ) = ( D n , n )
e = Δ
min n Q ( n ) , subject  to  K n d = Δ .
L μ ( n ) = 1 2 K n d 2 + μ 2 Q ( n ) ,
K T K n + μ D n K T d = 0.
i = 2 N 1 ( n i 1 2 n i + n i + 1 ) 2
D = [ 1 2 1 0 0 0 0 0 0 0 2 5 4 1 0 0 0 0 0 0 1 4 6 4 1 0 0 0 0 0 0 1 4 6 4 1 0 0 0 0 0 0 0 0 1 4 6 4 1 0 0 0 0 0 0 1 4 6 4 1 0 0 0 0 0 0 1 4 5 2 0 0 0 0 0 0 0 1 2 1 ] .
N = 200
K
W 1 , 2
W 1 , 2
n ( r )
τ aero
n ( r )
n ( r )
W 1 , 2
L 2
n e α ( e ) ( r ) W 1 , 2 [ a , b ]
e 0
n true ( r )
W 1 , 2 [ a , b ]
J α [ n ( r ) ] = ρ F [ K n , τ aero ] + α L ( n ) ,
ρ F [ K n , τ aero ] = 1 2 k ( r , λ , η ) n ( r ) τ aero ( λ ) L 2 2 ,
L ( n ) = 1 2 n ( r ) W 1 , 2 2 .
x ( t )
y ( t )
W 1 , 2
[ x ( t ) , y ( t ) ] w 1 , 2 = def Ω [ x ( t ) y ( t ) + i = 1 n x t i y t j × d t 1 d t 2 , …   , d t n ] ,
n ( r )
n ( r )
J α [ n ( r ) ]
α [ n ″( r ) n ( r ) ] a b k ¯ ( r , ξ , η ) n ( ξ ) = τ aero ¯ ( r ) ,
n ( a ) = 0 , n ( b ) = 0 ,
k ¯ ( r , ξ , η ) = a b k ( r , λ , η ) k ( ξ , λ , η ) d λ ,
τ aero ¯ ( r ) = a b k ( r , λ , η ) τ aero ( λ ) d λ .
n ( r )
n ( r )
{ r j } j = 1 N
A i j = { h r k ( r i , λ j , η ) , j = 2 , …   , N 1 h r 2 k ( r i , λ j , η ) , j = 1 , N
h r
h r = ( b a ) / ( N 1 )
h r
r i = a + ( i 0.5 ) h r , i = 1 , 2 , …   , N ; h r = b a N 1 ,
α [ 1 h r 2 ( n i 1 2 n i + n i + 1 ) n i ] j = 1 N h r k ¯ i j n j = τ aero i ¯ ,
i = 1 , 2 , …   , N 1 ,
n 1 n 0 h r = 0 , n N + 1 n N h r = 0 ,
n i = n ( r i ) ; k ¯ i j = k ¯ ( r i , ξ j , η )
τ aero i ¯ = τ aero ¯ ( r i )
k ¯ ( r i , ξ j , η ) = a b k ( r i , λ , η ) k ( ξ j , λ , η ) d λ ,
τ aero ¯ ( r i ) = a b k ( r i , λ , η ) τ aero ( λ ) d λ .
A = ( A i j ) N × N
n
τ aero ¯
A T A n + α H n A T τ aero ¯ = 0 ,
H = [ 1 + 1 / h r 2 - 1 / h r 2 0 0 - 1 / h r 2 1 + 2 / h r 2 - 1 / h r 2 0 0 - 1 / h r 2 1 + 2 / h r 2 - 1 / h r 2 0 0 - 1 / h r 2 1 + 1 / h r 2 ] .
[ 0.1 , 4 ]   μm
h r = 3.9 / ( N 1 )
N = 200
n ( r )
[ 0.1 , 10 ]   μm
( N 20 )
N = 200
h ( r )
f ( f ) : n ( r ) = h ( r ) f ( r )
h ( r )
f ( r )
τ aero ( λ ) = a b [ k ( r , λ , η ) h ( r ) ] f ( r ) d r ,
k ( r , λ , η ) = π r 2 Q ex ( r , λ , η )
k ( r , λ , η ) h ( r )
( Ξ f ) ( r ) = τ aero ( λ ) .
n ( r )
n α ( r )
n ( r )
α *
Ψ ( α ) = A n α τ aero ¯ 2 δ 2 ,
Ψ ( α )
α *
α k + 1 = α k 2 Ψ ( α k ) Ψ ( α k ) + [ Ψ ( α k ) 2 2 Ψ ( α k ) Ψ ″( α k ) ] 1 / 2 .
β ( α ) = H 1 / 2 n α 2 ,
Ψ ( α ) = α β ( α ) ,
Ψ ( α ) = β ( α ) 2 α [ H 1 / 2 d n α d α 2 + ( H n α , d 2 n α d α 2 ) ] ,
β ( α ) = 2 ( H d n α d α , n α ) .
n α
d n α / d α
d 2 n α / d α 2
( A T A + α H ) n α k = A T τ aero ¯ ,
( A T A + α H ) n α k = H n α k ,
( A T A + α H ) n α k = 2 H n α k .
A T A + α H
n α
d n α / d α
d 2 n α / d α 2
1.67 m / s
2.33 m / s
1.80 m / s
λ  ( μm )
η = 1.6 0.1 i
δ = 0.001
N = 200
α 0
α k
α *
α *
O ( 1.0 e 6 )
n ( r )
[ 0.4 , 0.7 ]   μm
[ 7 , 10 ]   μm
0.5   μm
7 10   μm
W 1 , 2
W 1 , 2
n ( r )
V 0
α *

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